Packing of chain segments: A method for describing X-ray patterns of crystalline, liquid crystalline and non-crystalline polymers

  • T. Pieper
  • H. -G. Kilian
Part of the Advances in Polymer Science book series (POLYMER, volume 108)


A general principle of chain segment packing is presented which covers a wide range of distance correlations occurring in macromolecular systems. The variety of local motions and defects in chain segments results in a delocalized, cylindrically symmetric electronic distribution along the chain. Thus, the molecules can be substituted for cylinder segments containing these conformationally averaged chain segments. The length of these cylinders is completely determined by the atom-to-atom distance range of the molecular autocorrelation function. The conformation elements obtained by appropriate statistical calculations were packed in the lateral direction by applying well-known principles from one-dimensional fluid models obeying the correct values of the mean site-to-site distance, distance fluctuations, exclusion volume, and macroscopic density.

Short range order in liquid-like systems as well as long range order in crystalline domains are reflected in WAXS-patterns very clearly. Some examples of calculated X-ray patterns from PTFE (Phase I), a smectic LC-phase and even a PE melt, show that our model covers a wide range of macromolecular structures running the whole scale from crystalline systems over mesophases up to polymer melts. The range of intra- and intermolecular order can be estimated fairly well with the help of density correlation functions.


Short Range Order Chain Segment Packing Fraction Distance Correlation WAXS Pattern 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Abbreviations and Symbols

\(\overrightarrow Q\)

scattering vector




Bragg angle

scattering angle

\(\hat \rho (\overrightarrow r )\)

density correlation function (DCF)


radial density difference function (RDDF)


distance distribution of next neighbours


radial distance distribution

\(\hat \rho _{auto} (\overrightarrow r )\)

auto-correlation function

\(\left\langle {\hat \rho _{auto} (\overrightarrow r )} \right\rangle\)

mean auto-correlation function


mean electron density

\(\left\langle {\rho (\overrightarrow r )} \right\rangle\)

mean molecular electron density


mean number density


lateral packing density (number of cylinders per unit area)

\(n_{CH_2 }\)

number density of CH2 units


number of CH2 units in correlation cylinder

npacking, nchain

radial and axial CH2 number density, respectively

Fk, F*1

molecular structure amplitudes

Iintra, L

isotropic average of mean intramolecular structure factor in laboratory system

Iinter, L

isotropic average of mean intermolecular structure factor in laboratory system

fi, fj

form factors of atoms i and j


mean molecular structure factor


mean molecular structure amplitude


Rotational Isomeric State Approximation


statistical weight of molecular conformation j


transition matrix


hard core diameter


mean distance


electronic radius of gyration


distance fluctuation parameter, \(g = \sqrt {\frac{{\left\langle {r^2 } \right\rangle }}{{\left\langle r \right\rangle ^2 }} - 1}\)


radial and axial component of \(\overrightarrow r\)

qr, qz or R, Z

radial and axial component of \(\overrightarrow Q\)


\(q = 2\pi \frac{{2sin\theta }}{\lambda }\)

\(\overrightarrow r _{kl}\)

distance vector between atoms k and l


space average


Bessel function of order n


angular distribution of atom j


Fourier expansion of gj(ϕ)

\(\bar M\)w

molecular weight


reduced scattering intensity


Avogadro number








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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • T. Pieper
    • 1
  • H. -G. Kilian
    • 1
  1. 1.Department of Experimental PhysicsUniversity of UlmUlmFRG

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